A few special problems of meromorphic functions share values in part with their shifts and derivatives

This article examines several uniqueness theorems for non-constant meromorphic functions of hyper-order strictly less than 1, whose k-th derivatives partially share values with their shifts on the extended complex plane. Additionally, our results build upon and enhance two earlier studies by Charak-Korhonen-Kumar (J. Math. Anal. Appl. \(435 (2):1241-1248, 2016\) $435(2):1241-1248,2016$ ) and Lin-Lin-Wu (Bull. Korean Math. Soc. \(55 (2):469-478, 2018\) $55(2):469-478,2018$ ). As a result, we demonstrate two uniqueness results for meromorphic functions of hyper-order strictly less than 1, whose k-th derivatives share four different values with shifts in the case of ignoring multiplicities. Also, some examples show how precise the parameters of our results were.